Let $\nabla$ be a connection on the tangent bundle of a manifold $M$ and let $\mathrm{\χ}\left(M\right)$ be the module of all vector fields on $M$. The curvature tensor of $\nabla$is the type $\left(1\,3\right)$ tensor $R$ (contravariant rank 1, covariant rank 3) which, when viewed as a linear map $R\:\mathrm{\χ}\left(M\right)\times \mathrm{\χ}\left(M\right)\to L\left(\mathrm{\χ}\left(M\right)\right)$, is given by

More generally, let $\nabla$ be a connection on a vector bundle $E\to M$ and let $\mathrm{\Σ}\left(E\right)$ be the module of all sections of $E$. The curvature tensor of $\nabla \,$ when viewed as a linear map $\mathrm{\Ω}\:\mathrm{\χ}\left(M\right)\times \mathrm{\χ}\left(M\right)\to L\left(\mathrm{\Σ}\left(E\right)\right)$, is given by

 The first calling sequence computes the Christoffel symbol of the input metric $g$ and returns the Riemann curvature tensor on the tangent bundle as a tensor of type $\left(\genfrac{}{}{0ex}{}{1}{3}\right)$. The index type of the output is ["con_bas", "cov_bas", "cov_bas", "cov_bas"]. See TensorIndexType.

The second calling sequence returns the curvature tensor of the input connection $C$. The index type of the output is ["con_vrt", "cov_vrt", "cov_bas", "cov_bas"]. See TensorIndexType.

The first Bianchi identity for the curvature tensor of a connection on the tangent bundle of a manifold asserts that

The second Bianchi Identity asserts that $\nabla R \+ R·S \left(\mathrm{skew}-\mathrm{symmetrized} \mathrm{on} \mathrm{indices} 3\,4\,5\right) \= 0$. Here $R·S$ denotes the contraction of $R\otimes S$ on the third and fifth indices.

This command is part of the DifferentialGeometry:-Tensor package, and so can be used in the form CurvatureTensor(...) only after executing the command with(DifferentialGeometry) and with(Tensor) in that order. It can always be used in the long form DifferentialGeometry:-Tensor:-CurvatureTensor.

$\mathrm{with}\left(\mathrm{DifferentialGeometry}\right)\:$$\mathrm{with}\left(\mathrm{Tensor}\right)\:$

$\mathrm{DGsetup}\left(\left[x\,y\,z\right]\,M\right)$

$\mathrm{frame\; name:\; M}$

$\mathrm{g1}≔\mathrm{evalDG}\left(\exp \left(y\right)\mathrm{dx}\&t\mathrm{dx}+\mathrm{dy}\&t\mathrm{dy}+\mathrm{dz}\&t\mathrm{dz}\right)$

$\mathrm{g1}:={\ⅇ}^y\mathrm{dx}\mathrm{dx}+\mathrm{dy}\mathrm{dy}+\mathrm{dz}\mathrm{dz}$

$\mathrm{R1}≔\mathrm{CurvatureTensor}\left(\mathrm{g1}\right)$

$\mathrm{R1}:=-\frac{1}{4}\mathrm{D\_x}\mathrm{dy}\mathrm{dx}\mathrm{dy}+\frac{1}{4}\mathrm{D\_x}\mathrm{dy}\mathrm{dy}\mathrm{dx}+\frac{{\ⅇ}^y}{4}\mathrm{D\_y}\mathrm{dx}\mathrm{dx}\mathrm{dy}-\frac{{\ⅇ}^y}{4}\mathrm{D\_y}\mathrm{dx}\mathrm{dy}\mathrm{dx}$

$\mathrm{DGsetup}\left(\left[x\,y\,z\right]\,M\right)$

$\mathrm{frame\; name:\; M}$

$\mathrm{C2}≔\mathrm{Connection}\left(x^2\left(\mathrm{D\_x}\&t\mathrm{dx}\right)\&t\mathrm{dy}-y^2\left(\mathrm{D\_x}\&t\mathrm{dy}\right)\&t\mathrm{dy}+yz\left(\mathrm{D\_x}\&t\mathrm{dz}\right)\&t\mathrm{dy}\right)$

$\mathrm{C2}:=x^2\mathrm{D\_x}\mathrm{dx}\mathrm{dy}-y^2\mathrm{D\_x}\mathrm{dy}\mathrm{dy}+yz\mathrm{D\_x}\mathrm{dz}\mathrm{dy}$

$\mathrm{R2}≔\mathrm{CurvatureTensor}\left(\mathrm{C2}\right)$

$\mathrm{R2}:=2x\mathrm{D\_x}\mathrm{dx}\mathrm{dx}\mathrm{dy}-2x\mathrm{D\_x}\mathrm{dx}\mathrm{dy}\mathrm{dx}-y\mathrm{D\_x}\mathrm{dz}\mathrm{dy}\mathrm{dz}+y\mathrm{D\_x}\mathrm{dz}\mathrm{dz}\mathrm{dy}$

Bianchi1 := proc(R, C)

$\mathbf{local} S\,\mathrm{T1}\,\mathrm{T2}\,\mathrm{T3}\,\mathrm{T4}$

$S≔\mathrm{TorsionTensor}\left(C\right)$

$\mathrm{T1}≔R$

$\mathrm{T2}≔\mathrm{CovariantDerivative}\left(S\,C\right)\:$

$\mathrm{T3}≔\mathrm{ContractIndices}\left(S\,S\,\left[\left[3\,1\right]\right]\right)$

$\mathrm{T4}≔\mathrm{evalDG}\left(\mathrm{T1}-\mathrm{T2}+\mathrm{T3}\right)$

$\mathrm{SymmetrizeIndices}\left(\mathrm{T4}\,\left[2\,3\,4\right]\,''SkewSymmetric''\right)$

end:

Bianchi2 := proc(R, C)

$\mathbf{local} \mathrm{T1}\,\mathrm{T2}\,S$

$S≔\mathrm{TorsionTensor}\left(C\right)$

$\mathrm{T1}≔\mathrm{CovariantDerivative}\left(R\,C\right)$

$\mathrm{T2}≔\mathrm{ContractIndices}\left(R\,S\,\left[\left[3\,1\right]\right]\right)$

$\mathrm{SymmetrizeIndices}\left(\mathrm{T1}\&plus\mathrm{T2}\,\left[3\,4\,5\right]\,''SkewSymmetric''\right)$

end:

$\mathrm{Bianchi1}\left(\mathrm{R2}\,\mathrm{C2}\right)$

$0\mathrm{D\_x}\mathrm{dx}\mathrm{dx}\mathrm{dx}$

$\mathrm{Bianchi2}\left(\mathrm{R2}\,\mathrm{C2}\right)$

$0\mathrm{D\_x}\mathrm{dx}\mathrm{dx}\mathrm{dx}\mathrm{dx}$

$\mathrm{DGsetup}\left(\left[x\,y\,z\right]\,M\right)$

$\mathrm{frame\; name:\; M}$

$\mathrm{FR}≔\mathrm{FrameData}\left(\left[\frac{x^2}{y}\mathrm{dx}\,\frac{z}{x}\mathrm{dy}\,xy\mathrm{dz}\right]\,\mathrm{M1}\right)\:$

$\mathrm{DGsetup}\left(\mathrm{FR}\right)$

$\mathrm{frame\; name:\; M1}$

$\mathrm{C3}≔\mathrm{Connection}\left(\left(\mathrm{E2}\&t\mathrm{\Θ1}\right)\&t\mathrm{\Θ2}\right)$

$\mathrm{C3}:=\mathrm{E2}\mathrm{\Θ1}\mathrm{\Θ2}$

$\mathrm{R3}≔\mathrm{CurvatureTensor}\left(\mathrm{C3}\right)$

$\mathrm{R3}:=-\frac{y}{x^3}\mathrm{E2}\mathrm{\Θ1}\mathrm{\Θ1}\mathrm{\Θ2}+\frac{y}{x^3}\mathrm{E2}\mathrm{\Θ1}\mathrm{\Θ2}\mathrm{\Θ1}-\frac{1}{zxy}\mathrm{E2}\mathrm{\Θ1}\mathrm{\Θ2}\mathrm{\Θ3}+\frac{1}{zxy}\mathrm{E2}\mathrm{\Θ1}\mathrm{\Θ3}\mathrm{\Θ2}$

$\mathrm{Bianchi1}\left(\mathrm{R3}\,\mathrm{C3}\right)$

$0\mathrm{E1}\mathrm{\Θ1}\mathrm{\Θ1}\mathrm{\Θ1}$

$\mathrm{Bianchi2}\left(\mathrm{R3}\,\mathrm{C3}\right)$

$0\mathrm{E1}\mathrm{\Θ1}\mathrm{\Θ1}\mathrm{\Θ1}\mathrm{\Θ1}$

$\mathrm{DGsetup}\left(\left[x\,y\,z\right]\,\left[u\,v\,w\right]\,E\right)$

$\mathrm{frame\; name:\; E}$

$\mathrm{C4}≔\mathrm{Connection}\left(xz\left(\mathrm{D\_v}\&t\mathrm{du}\right)\&t\mathrm{dy}\right)$

$\mathrm{C4}:=xz\mathrm{D\_v}\mathrm{du}\mathrm{dy}$

$\mathrm{R4}≔\mathrm{CurvatureTensor}\left(\mathrm{C4}\right)$

$\mathrm{R4}:=z\mathrm{D\_v}\mathrm{du}\mathrm{dx}\mathrm{dy}-z\mathrm{D\_v}\mathrm{du}\mathrm{dy}\mathrm{dx}-x\mathrm{D\_v}\mathrm{du}\mathrm{dy}\mathrm{dz}+x\mathrm{D\_v}\mathrm{du}\mathrm{dz}\mathrm{dy}$